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Development of a Model for Applying Correlation Cryptanalysis to Filtering Generators

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Development of a Model for Applying Correlation Cryptanalysis to Filtering Generators NOVATEUR PUBLICATIONS JournalNX- A Multidisciplinary Peer Reviewed Journal ISSN No: 2581 - 4230 VOLUME 6, ISSUE 5, May -2020 DEVELOPMENT OF A MODEL FOR APPLYING CORRELATION CRYPTANALYSIS TO FILTERING GENERATORS ABDURAKHIMOV BAKHTIYOR FAYZIEVICH Professor, Doctor of Physics and Mathematics, National university of Uzbekistan, +998935143137, [email protected] BOYKUZIEV ILKHOM MARDANAKULOVICH PhD Student, National university of Uzbekistan, +998909779300, [email protected] SHONAZAROV SOATMUROD KULMURODOVICH Teacher, Termez state university, +998996760166, [email protected] ZIYAKULOVA SHAKHNOZA ABDURASULOVNA Teacher, Termez state university, +998905208923, [email protected] ABSTRACT: Proposed mathematical model and This article is devoted to problems of software can be used for estimation of cryptanalysis. Cryptographic analysis stability of the algorithm of the flow crypto results, got from using of correlation operation to correlation cryptanalysis, as cryptanalysis method to algorithm of the well as in scholastic purpose. flow crypto operation, founded on register of KEYWORDS: cryptanalysis, stream ciphers, the shift with feedback, were presented in shift registers, LILI-128, filtering, work. For estimation of the flow crypto correlation operation algorithm’s crypto stability by correlation cryptanalysis method, number INTRODUCTION: of important tasks was defined, purposes and problems of the study were determined. Ensuring information security requires As a result of defined tasks decision, relatively fast cryptographic tools, not only as mathematical model and software of the exchange of documentary information correlation cryptanalysis method to flow transmitted over the network increases, but crypto operation to filtering LILI-128 – were also as the exchange of multimedia, that is, video developed. Results has shown, that and audio, increases. Therefore, the use of characteristic of correlation immunity of stream encryption algorithms in local and global filtering functions, used in filtering networks has become an urgent problem[1,2]. generator, provide stability to correlation When analyzing stream encryption cryptanalysis method. Execution with big algorithms, in contrast to block encryption probability of stat-analogical function, algorithms, although many original ideas and defined in process of cryptanalysis, raises directions for creating stream encryption the efficiency of cryptanalysis. cryptographic algorithms were developed in this area, there is no single way to express270 | Page their commonality[3]. NOVATEUR PUBLICATIONS JournalNX- A Multidisciplinary Peer Reviewed Journal ISSN No: 2581 - 4230 VOLUME 6, ISSUE 5, May -2020 Therefore, it is important to conduct research in the field of continuous encryption algorithms and modern methods for assessing their cryptographic strength. Due to the design features of continuous encryption algorithms, the most common type of attack is a correlation attack. If a non-linear function transfers information about its internal components to the output, the work required to open such a system is greatly reduced. However, such a function is always available[4]. According to this axiom, correlation attacks use the correlation between a sequence leaving an st nd Figure 1. Scheme of 1 and 2 generators encryption scheme and a sequence leaving i1 i2 ik l(x) = x ⊕ x ⊕ … ⊕ x ⊕ b depending registers[5,6,7]. METHODOLOGY: on the value of the free variable b (b =0 ва b = 1)nd 2.0 Correlation cryptanalysis method for in linear stanology, the operation of the 2 filter generators generator can be expressed using one of the schemes shown in the figure 2. There are several methods of correlation cryptanalysis applied to continuous encryption algorithms based on filter generators. Underlying these approaches is the use of linear statanalogs of the filtering function. Suppose that in a filter generator there is a linear feedback shift register of length n, 1 n n 1 1 n φ(x ,...,x ) = c x ⊕ … ⊕ c x is the feedback n n 1 function of the register, C(D) = c D ⊕ … ⊕ c D nd ⊕ 1 is its characteristic multiplication, and f 1 n Figure 2. Operating∞ status of 2 generator (x ,...,x ) is a filtering function, let the linear ij i1 i2 The series S } are different recurrence statanology of the function f(x) be l(x) = x ⊕ x ik sequences formed by one shift register with a 1 n ⊕ … ⊕ x ⊕ b . The probability that the feedback function φ(x ,...,x ), length n. From the function f (x) and its statonology coincide with known results on the linear complexity of the l(x) is: Prob{f(x) = l(x)} = 1/2 + a/2 [5]. Where sum of linear recurrence sequences and the – is the maximum value of linear complexity of inverted sequences, it can the normalized modulus of the Fourierst be seen that the length of the sequence {y’(t)} is coefficient. We call this given generator 1 n, and the feedback function is still equal to 1 n n 1 1 n generator. Filter generator with a similar shift φ(x ,...,x ) = c x ⊕ … ⊕ c x or length n + 1 and register, but with a filter functionnd l(x). This multiplication of the characteristic n n+1 n n-1 n n-1 n-2 n- generator is called the 2 generator. The C’(D) = c D ⊕(c ⊕c )D ⊕(c ⊕c )D 1 2 1 2 1 following symbols {y(t)} and {y’(t)} denote the ⊕…⊕(c ⊕c )D ⊕(c ⊕1)D⊕1 (1) i output sequences generated by the generators 1 which is made using offset sizes. Where c is the 1 n and 2, respectively, when the initial filled state coefficient of the feedback φ(x ,...,x ) function. is the same (Figure 1). 271 | Page NOVATEUR PUBLICATIONS JournalNX- A Multidisciplinary Peer Reviewed Journal ISSN No: 2581 - 4230 VOLUME 6, ISSUE 5, May -2020 Thus, instead of the 2nd generator, we can take before. In this case, therd bits of the output a look at the equivalent 3-generator (Figure 3). sequence {y’(t)} of the 3 generator correspond Therefore, in the case b = 0, the register length to the bits of the-m sequence {y(t)} with a is L=n, and the feedback function is φ’(x) = φ(x), probability of 1 – 2 . In this case, assuming that and in the case b = 1, the register length is all 2n bits of the sequence {y(t)} correspond to L=n+1, and the coefficient of the feedback bits of the sequence {y’(t)}, we can construct rda function C’(D) is determined from the feedback linear displacement registerst (3 characteristic coefficient of thend polynomial. generator) at the output of the 1 generator. Then rdthe output sequences of 2 generators Then this generator is used as a decoder with-m and 3 generators coincide and are equal to the correct decryption probability of 1 – 2 . {y’(t)}, and then the following equation holds forst Therefore, if the function f(x) hasst certain the elements ofnd the output sequences of 1 properties, the operation of the 1 generator generators and 2 generators: Prob{ y(t)= y’(t)} can be completely restored, that is, the unknown = 1/2 + a/2. feedback function and the initial state of the register can be determined. As a rule, to assess the reliability of the continuous encryption algorithm, the cryptanalyst is informed about the gamma sequence and internal structure of the generator, which come from a generator of a certain length. That is, the length of the register used in the generator, the feedback, and the Figure 3. Scheme of generator 3 appearance of the filter function are known. If the 1st generator is used to encrypt Based on this, using the two above some “ordinary” information and the parameter approaches, the method of correlation а is large enough, the 3rd generator can be used cryptanalysis is applied to the filter generator as Example 1. as a decoder, and errors can be eliminated follows. depending on the capabilities of the language. Let the above 1 be a Consider a situation that has the form: generator (Figure 4). It is known that this The feedback function of the above approach generator has an offset register of length 7 and st 1 n the 1 generator φ(x ,...,x ) is unknown, and the a filter function f(x) as follows: 1 1 2 3 4 5 6 7 filter function f(x) is as follows: f(x ,...,xn) = f(x) = x ⊕ x * x * x * x * x ⊕ x 1 2 m m+1 m+2 n. x *x *…*x ⊕ x ⊕ x ⊕…⊕ x The φ feedback function is as follows: 1 7 1 2 3 6 Where m ≤ n, n is the length of the displacement φ(x ,…,x ) = x ⊕ x ⊕ x ⊕ x dimensions. The linear statonology of the The initial filled state of the generator is m+1 n m+1 function f(x) is the function l(x ,…,x ) = x ⊕ unknown. m+2 n x ⊕…⊕ x with the probability-m of coincidence Prob{f(x) = l(x)} = 1 - 2 . In particular, for m ≥ 5, the probability is Prob{f(x) = l(x)} > 0,969. st Thus, if the filtering function f(x) in a 1 generator is replacednd by its linear statistical analog,rd the resulting 2 generator is equivalent to a 3 generator, i.e. there is an n-length shift register with the same feedback function as Figure 4. Filtering generator272 | Page NOVATEUR PUBLICATIONS JournalNX- A Multidisciplinary Peer Reviewed Journal ISSN No: 2581 - 4230 VOLUME 6, ISSUE 5, May -2020 nd However, at least 14 consecutive bits of The output bits of the 2 generator the output sequence {y(t)} are specified, let be: sequence y’(t) are calculated using a linear filter 1 7 00101001110010. The purpose of the analysis l(x) = x ⊕ x . is to find the initial state of the register. Thus, the following system of equations The linear statanology of the filter (4) is suitable: 1 7 function f(x) is the function l(x) = x ⊕ x .
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